Quick answer
Polynomial long division divides one polynomial by another. (x² − 3x + 2) ÷ (x − 1) = x − 2 with remainder 0, entered as coefficients [1, −3, 2] and [1, −1].
Enter the dividend and divisor coefficients from the highest power. The calculator returns the quotient and remainder.
Related tools: Polynomial Calculator: Add, subtract or multiply two polynomials, and evaluate the result. Long Division Calculator: Divide whole numbers to a chosen number of decimal places. Slope Calculator: Slope, angle, distance and line equation through two points.
How to use the Polynomial Long Division Calculator
- Type dividend coefficients (highest power first), for example 1, -3, 2.
- Type divisor coefficients (highest power first), for example 1, -1.
- Press Calculate to see the result, and use Reset to start again.
| Input | What to enter | Allowed values |
|---|---|---|
| Dividend coefficients (highest power first) | Text, such as 1, -3, 2 | — |
| Divisor coefficients (highest power first) | Text, such as 1, -1 | — |
Formula and method
Formula
long division of polynomials
IEEE-754 double precision; shown to the decimals indicated.
Coefficients are numbers; results are shown as decimals.
Worked examples
These examples are verified: each one is an automated test case that the live calculator must reproduce exactly.
Example 1
Inputs: Dividend coefficients (highest power first) = 1, -3, 2; Divisor coefficients (highest power first) = 1, -1.
| Result | Value |
|---|---|
| Quotient | 1, -2 |
| Remainder | 0 |
Example 2
Inputs: Dividend coefficients (highest power first) = 2, 0, -5, 3; Divisor coefficients (highest power first) = 1, 1.
| Result | Value |
|---|---|
| Quotient | 2, -2, -3 |
| Remainder | 6 |
The first line is the headline result (quotient). Below it the polynomial Long Division Calculator lists remainder, which describe the same case from other angles. IEEE-754 double precision; shown to the decimals indicated.
Steps
x² ÷ x = x; x(x − 1) = x² − x; subtract to get −2x + 2; −2x ÷ x = −2; −2(x − 1) = −2x + 2; remainder 0. So x² − 3x + 2 = (x − 1)(x − 2).
Factors and remainders
A zero remainder means the divisor is a factor, and x = 1 is a root. The remainder theorem says dividing by (x − c) leaves f(c). The leading divisor coefficient must not be zero.
What happens with invalid input
These messages are produced by the calculator itself, so you know what to fix.
- Dividend coefficients (highest power first): left empty shows "Enter dividend coefficients (highest power first)." beside that field and no result.
- Divisor coefficients (highest power first): entering "0, 1" shows "The leading coefficient of the divisor cannot be zero." beside that field and no result.
Choosing between the Polynomial Long Division Calculator and similar calculators
Each row shows what a related calculator on this site asks for and what it returns, so you can pick the one that matches your question.
| Calculator | You enter | It returns |
|---|---|---|
| [Polynomial Calculator](/math/polynomial-calculator/) | first polynomial coefficients, second polynomial coefficients and operation | Result coefficients |
| [Long Division Calculator](/math/long-division-calculator/) | dividend and divisor | Quotient |
| [Slope Calculator](/math/slope-calculator/) | x₁, y₁ and x₂ | Slope |
| [LCM Calculator](/math/lcm-calculator/) | numbers | Least common multiple |
| [Factor Calculator](/math/factor-calculator/) | number | Factors |
Background: algebra
Many algebra calculators rest on the same ideas: write the problem in a standard form, identify the coefficients, apply the formula and check the answer by substituting it back. For a quadratic ax² + bx + c = 0 the roots are (−b ± √(b² − 4ac)) ÷ 2a, and the discriminant b² − 4ac decides the kind of roots: positive gives two real roots, zero gives one repeated root and negative gives two complex conjugate roots.
If a = 0 the equation is linear, not quadratic. A sign slip on the −b term shifts both roots by the same amount, so substitute your roots back to check.
Assumptions and limitations
- Leading divisor coefficient must not be zero
If a required value is empty, not a number, or outside its allowed range, the form marks that field with a message and shows no result instead of a misleading number. The automated checks deliberately try invalid entries for dividend coefficients (highest power first) and divisor coefficients (highest power first) to confirm this.
Frequently asked questions
What is the remainder of x² + 1 divided by x − 1?
By the remainder theorem, dividing x² + 1 by x − 1 leaves f(1) = 2.
Can the divisor have higher degree?
Then the quotient is zero.
What is synthetic division?
A shortcut for linear divisors.
What does the Polynomial Long Division Calculator calculate?
Quotient and remainder of one polynomial divided by another. It returns Quotient and Remainder.
What formula does the Polynomial Long Division Calculator use?
long division of polynomials. Coefficients are numbers; results are shown as decimals.
What do I need to enter in the polynomial Long Division Calculator?
You enter Dividend coefficients (highest power first) and Divisor coefficients (highest power first).
How is the result rounded or limited?
IEEE-754 double precision; shown to the decimals indicated. Leading divisor coefficient must not be zero.
What is an example of a polynomial Long Division result?
With dividend coefficients (highest power first) 1, -3, 2 and divisor coefficients (highest power first) 1, -1, the quotient is 1, -2.
Related calculators
- Polynomial Calculator — Add, subtract or multiply two polynomials, and evaluate the result.
- Long Division Calculator — Divide whole numbers to a chosen number of decimal places.
- Slope Calculator — Slope, angle, distance and line equation through two points.
- LCM Calculator — Least common multiple of two or more numbers.
- Factor Calculator — List every factor of a positive whole number.
- Quadratic Formula Calculator — Solve ax² + bx + c = 0, including complex roots.
Sources
- OpenStax, Prealgebra 2e (Rice University): fractions, decimals, percents, ratios and rounding
- NIST Special Publication 811, Guide for the Use of the International System of Units (SI)
Content last reviewed 2026-11-04. Formulas and worked examples are checked by automated tests against independently computed values.